I came across the following technical question, to which I could not - after some time of thinking - find an answer:
Let $\mathcal{U},\mathcal{H}$ be two real (in general infinite dimensional) separable Hilbert spaces. For some linear subspace $\bar{U} \subseteq \mathcal{U}$, let $\Pi_U$ denote the orthogonal projection on this subspace. The question is:
Is the mapping $T \mapsto \Pi_{\text{ker}T}$ measurable from $L(\mathcal{U},\mathcal{H})$ to $L(\mathcal{U})$ when both spaces are endowed with the strong operator topology (and measurability is meant w.r.t. the Borel-$\sigma$-algebra of the SOT on both sides)?
Any hints and thoughts on this are more than appreciated!